An elementary introduction to the geometry of

quantum states with a picture book

J. Avron, O. Kenneth

Dept. of Physics, Technion, Israel

January 23, 2019

Abstract

This is a review of the geometry of quantum states using elementary

methods and pictures. Quantum states are represented by a convex body,

often in high dimensions. In the case ofn-qubits, the dimension is expo-

nentially large inn. The space of states can be visualized, to some extent,

by its simple cross sections: Regular simplexes, balls and hyper-octahedra^1.

When the dimension gets large there is a precise sense in which the space of

states resembles, almost in every direction, a ball. The ball turns out to be

a ball of rather low purity states. We also address some of the correspond-

ing, but harder, geometric properties of separable and entangled states and

entanglement witnesses..

“All convex bodies behave a bit like Euclidean balls.”

Keith Ball

Contents

1 Introduction 2

1.1 The geometry of quantum states.................... 2

1.2 The geometry of separable states................... 6

2 Two qubits 8

2.1 Numerical sections for 2 qubits.................... 8

2.2 A 3-D section through Bell states................... 9

(^1) Also known as cross polytope, orthopex, and co-cube

arXiv:1901.06688v1 [quant-ph] 20 Jan 2019